Generalized Maxwell Model — Canonical Equations

This document collects the canonical equations for the Generalized Maxwell Model (GMM / Prony series) as used in linear viscoelasticity. All equations below are standard results derivable from the constitutive relation of a parallel arrangement of N Maxwell elements plus an equilibrium spring.

References

• Ferry, J. D. (1980). Viscoelastic Properties of Polymers, 3rd ed. Wiley.

• Tschoegl, N. W. (1989). The Phenomenological Theory of Linear Viscoelastic Behavior. Springer.

• Baumgaertel, M. & Winter, H. H. (1989). Determination of discrete relaxation and retardation time spectra from dynamic mechanical data. Rheol. Acta, 28, 511–519.

• Honerkamp, J. & Weese, J. (1989). Determination of the relaxation spectrum by a regularization method. Macromolecules, 22(11), 4372–4377.

• Park, S. W. & Schapery, R. A. (1999). Methods of interconversion between linear viscoelastic material functions. Part I. Int. J. Solids Struct., 36, 1653–1675.

1. Relaxation Modulus — Prony Series

The N-mode discrete relaxation spectrum:

\[G(t) = G_e + \sum_{i=1}^{N} G_i \, \exp\!\left(-\frac{t}{\tau_i}\right)\]

where:

• \(G_e \geq 0\) is the equilibrium (rubbery) modulus. For liquids, \(G_e = 0\); for crosslinked solids \(G_e > 0\).

• \(G_i > 0\) are the relaxation strengths (mode amplitudes).

• \(\tau_i > 0\) are the relaxation times.

• \(G_0 = G_e + \sum_i G_i\) is the instantaneous (glassy) modulus at \(t = 0^+\).

2. Storage Modulus

Obtained by one-sided Fourier cosine transform of \(G(t)\):

\[G'(\omega) = G_e + \sum_{i=1}^{N} G_i \, \frac{\omega^2 \tau_i^2} {1 + \omega^2 \tau_i^2}\]

Limiting behavior:

• \(\omega \to 0\): \(G'(\omega) \to G_e\)

• \(\omega \to \infty\): \(G'(\omega) \to G_0 = G_e + \sum_i G_i\)

3. Loss Modulus

Obtained by one-sided Fourier sine transform of \(G(t)\):

\[G''(\omega) = \sum_{i=1}^{N} G_i \, \frac{\omega \tau_i} {1 + \omega^2 \tau_i^2}\]

Each mode contributes a Debye peak at \(\omega = 1/\tau_i\) with maximum value \(G_i / 2\).

4. Complex Modulus and Complex Viscosity

The complex modulus is:

\[G^*(\omega) = G'(\omega) + i\, G''(\omega) = G_e + \sum_{i=1}^{N} G_i \, \frac{i\omega\tau_i} {1 + i\omega\tau_i}\]

The complex viscosity \(\eta^*(\omega) = G^*(\omega) / (i\omega)\):

\[\eta^*(\omega) = \sum_{i=1}^{N} \frac{G_i \tau_i}{1 + i\omega\tau_i} + \frac{G_e}{i\omega}\]

For liquids (\(G_e = 0\)):

\[\eta^*(\omega) = \sum_{i=1}^{N} \frac{G_i \tau_i}{1 + i\omega\tau_i}\]

Components:

\[\eta'(\omega) = \sum_{i=1}^{N} \frac{G_i \tau_i}{1 + \omega^2 \tau_i^2}\]

\[\eta''(\omega) = \frac{G_e}{\omega} + \sum_{i=1}^{N} \frac{G_i \omega \tau_i^2}{1 + \omega^2 \tau_i^2}\]

5. Creep Compliance

The creep compliance \(J(t)\) is the response to a step stress \(\sigma_0\). For the GMM, \(J(t)\) is the inverse Laplace transform relation to \(G(t)\):

\[s\,\hat{G}(s)\,s\,\hat{J}(s) = 1\]

where \(\hat{G}(s)\) and \(\hat{J}(s)\) are the Laplace transforms.

Exact retardation spectrum (Tschoegl 1989, Ch. 4):

For an N-mode GMM with \(G_e > 0\), the creep compliance takes the retardation form:

\[J(t) = J_g + \sum_{j=1}^{N} J_j \left[1 - \exp\!\left(-\frac{t} {\lambda_j}\right)\right]\]

where \(J_g = 1/G_0\) is the glassy compliance, \(\lambda_j\) are the retardation times (roots of the polynomial \(\sum_i G_i \prod_{k \neq i} (1 + s\tau_k) = 0\)), and \(J_j\) are the retardation strengths.

For liquids (\(G_e = 0\)), an additional viscous flow term appears:

\[J(t) = J_g + \sum_{j=1}^{N-1} J_j \left[1 - \exp\!\left(-\frac{t} {\lambda_j}\right)\right] + \frac{t}{\eta_0}\]

Numerical approach (backward-Euler, Park & Schapery 1999):

The GMM constitutive ODE for each mode under creep loading \(\sigma(t) = \sigma_0 H(t)\) is:

\[\frac{d\sigma_i}{dt} = -\frac{\sigma_i}{\tau_i} + G_i \dot\varepsilon\]

with total stress balance \(\sigma_0 = G_e\,\varepsilon + \sum_i \sigma_i\). The backward-Euler update uses exponential integration for unconditional stability:

\[\alpha_i = \exp(-\Delta t / \tau_i), \quad \beta_i = G_i \tau_i (1 - \alpha_i) / \Delta t\]

\[\Delta\varepsilon = \frac{\sigma_0 - \sum_i \alpha_i \sigma_i^n} {G_e + \sum_i \beta_i}\]

6. Zero-Shear Viscosity

\[\eta_0 = \int_0^\infty G(t)\,dt = \sum_{i=1}^{N} G_i \tau_i\]

This equals \(\lim_{\omega\to 0} \eta'(\omega)\) and is well-defined only for liquids (\(G_e = 0\)).

7. Stress Growth Coefficient (Startup Flow)

Under constant shear rate \(\dot\gamma\), the stress growth coefficient is:

\[\eta^+(t) = \frac{\sigma(t)}{\dot\gamma} = \sum_{i=1}^{N} G_i \tau_i \left[1 - \exp\!\left(-\frac{t} {\tau_i}\right)\right]\]

with \(\lim_{t\to\infty} \eta^+(t) = \eta_0\).

8. Fitting Methods for Spectrum Identification

Non-Negative Least Squares (NNLS):

Given fixed \(\tau_i\) on a log-spaced grid, solve for \(G_i \geq 0\):

\[\min_{G_i \geq 0} \left\| \mathbf{y} - \mathbf{A}\,\mathbf{g} \right\|^2\]

where \(A_{kj} = \exp(-t_k/\tau_j)\) for relaxation or the appropriate kernel for \(G'(\omega)\) / \(G''(\omega)\).

Tikhonov Regularization (Honerkamp & Weese 1989):

\[\min_{\mathbf{g}} \left\| \mathbf{y} - \mathbf{A}\,\mathbf{g} \right\|^2 + \alpha \left\| \mathbf{L}\,\mathbf{g} \right\|^2\]

where \(\mathbf{L}\) is typically a first- or second-order difference operator and \(\alpha > 0\) is the regularization parameter (chosen via L-curve, GCV, or Bayesian evidence).

Maximum Entropy (MEM):

\[\min_{\mathbf{g}} \left\| \mathbf{y} - \mathbf{A}\,\mathbf{g} \right\|^2 - \alpha \sum_j \left[ g_j \ln\!\left(\frac{g_j}{m_j}\right) - g_j + m_j \right]\]

where \(m_j\) is a default model (flat prior). MEM naturally enforces \(g_j > 0\) and produces the smoothest spectrum consistent with the data.

Two-Step NLSQ with Softmax Penalty (RheoJAX approach):

Step 1: Fit with differentiable penalty encouraging \(G_i > 0\):

\[P(\mathbf{G}) = s \sum_i \ln\!\left(1 + \exp(-G_i/s)\right)\]

Step 2: If any \(G_i < 0\), refit with hard bounds \(G_i \geq 0\).

Element Minimization (Baumgaertel & Winter 1989):

Start from large N, iteratively reduce to find smallest N where \(R^2 \geq R^2_\text{threshold}\). Uses warm-start from N+1 solution.

9. Discrete vs. Continuous Relaxation Spectrum

The continuous spectrum \(H(\tau)\) generalizes the discrete Prony series:

\[G(t) = G_e + \int_0^\infty H(\tau)\,\exp\!\left(-\frac{t}{\tau}\right) \,\frac{d\tau}{\tau}\]

\[G'(\omega) = G_e + \int_0^\infty H(\tau)\,\frac{\omega^2\tau^2} {1 + \omega^2\tau^2}\,\frac{d\tau}{\tau}\]

\[G''(\omega) = \int_0^\infty H(\tau)\,\frac{\omega\tau} {1 + \omega^2\tau^2}\,\frac{d\tau}{\tau}\]

The discrete spectrum is formally:

\[H(\tau) = \sum_{i=1}^{N} G_i\,\tau_i\,\delta(\tau - \tau_i)\]

Recovering \(H(\tau)\) from data is an ill-posed inverse problem, motivating the regularization approaches in Section 8.

The retardation spectrum \(L(\lambda)\) plays the analogous role for creep compliance:

\[J(t) = J_g + \int_0^\infty L(\lambda)\left[1 - \exp\!\left(-\frac{t} {\lambda}\right)\right]\frac{d\lambda}{\lambda} + \frac{t}{\eta_0}\]

10. Interconversion Identities

Key relationships between material functions (Ferry 1980, Ch. 3-4):

\[G'(\omega) = \omega \int_0^\infty G(t)\,\sin(\omega t)\,dt\]

\[G''(\omega) = \omega \int_0^\infty G(t)\,\cos(\omega t)\,dt\]

\[|G^*(\omega)| = \sqrt{G'^2 + G''^2}\]

\[\tan\delta = \frac{G''}{G'}\]

\[|\eta^*(\omega)| = \frac{|G^*(\omega)|}{\omega}\]

The Cox-Merz rule (empirical, for linear polymers):

\[|\eta^*(\omega)| \approx \eta(\dot\gamma)\big|_{\dot\gamma = \omega}\]